Optimal estimation of transducer parameters

ABSTRACT

The invention relates to an arrangement and method for estimating the linear and nonlinear parameters of a model  11  describing a transducer  1  which converts input signals x(t) into output signals y(t) (e.g., electrical, mechanical or acoustical signals). Transducers of this kind are primarily actuators (loudspeakers) and sensors (microphones), but also electrical systems for storing, transmitting and converting signals. The model describes the internal states of the transducer and the transfer behavior between input and output both in the small- and large-signal domain. This information is the basis for measurement applications, quality assessment, failure diagnostics and for controlling the transducer actively. The identification of linear and nonlinear parameters P l  and P n  of the model without systematic error (bias) is the objective of the current invention. This is achieved by using a transformation system  55  to estimate the linear parameters P l  and the nonlinear parameters P n  with separate cost functions.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to an arrangement and a method forestimating the linear and nonlinear parameters of a model describing atransducer which converts input signals (e.g., electrical, mechanical oracoustical signals) into output signals (e.g., electrical, mechanical oracoustical signals). Transducers of this kind are primarily actuators(such as loudspeakers) and sensors (such as microphones), but alsoelectrical systems for storing, transmitting and converting signals. Themodel is nonlinear and describes the internal states of the transducerand the transfer behavior between input and output at small and highamplitudes. The model has free parameters which have to be identifiedfor the particular transducer at high precision while avoiding anysystematic error (bias). The identification of nonlinear systems is thebasis for measurement applications, quality assessment and failurediagnostics and for controlling the transducer actively.

2. Description of the Related Art

Most of the nonlinear system identification techniques known in priorart are based on generic structures such as polynomial filters using theVolterra-Wiener-series as described by V. J. Mathews, AdaptivePolynomial Filters, IEEE SP MAGAZINE, July 1991, pages 10-26. Thosemethods use structures with sufficient complexity and a large number offree parameters to model the real system with sufficient accuracy. Thisapproach is not applicable to an electro-acoustical transducer as thecomputational load can not be processed by available digital signalprocessors (DSPs). However, by exploiting a priori information onphysical relationships it is possible to develop special modelsdedicated to a particular transducer as disclosed in U.S. Pat. No.5,438,625 and by J. Suykens, et al., “Feedback linearization ofNonlinear Distortion in Electro-dynamic Loudspeakers,” J. Audio Eng.Soc., 43, pp 690-694). Those models have a relatively low complexity anduse a minimal number of states (displacement, current, voltage, etc.)and free parameters (mass, stiffness, resistance, inductance, etc.).Static and dynamic methods have been developed for measuring theparameters of those transducer-oriented models. The technique disclosedby W. Klippel, “The Mirror Filter—a New Basis for Reducing NonlinearDistortion Reduction and Equalizing Response in Woofer Systems”, J.Audio Eng. Society 32 (1992), pp. 675-691, is based on a traditionalmethod for measuring nonlinear distortion. The excitation signal is atwo-tone signal generating sparse distortion components which can beidentified as harmonic, summed-tone or difference tone components of acertain order. This method is time consuming and can not be extended toa multi-tone stimulus because the distortion components interfere if thenumber of fundamental tones is high. In order to estimate the nonlinearparameters with an audio-like signal (e.g., music), adaptive methodshave been disclosed in DE 4332804A1 or W. Klippel, “Adaptive NonlinearControl of Loudspeaker Systems,” J Audio Eng. Society 46, pp. 939-954(1998).

Patents DE 4334040, WO 97/25833, US 2003/0118193, U.S. Pat. No.6,269,318 and U.S. Pat. No. 5,523,715 disclose control systems based onthe measurement of current and voltage at the loudspeaker terminalswhile dispensing with an additional acoustical or mechanical sensor.

Other identification methods, such as those disclosed in U.S. Pat. No.4,196,418, U.S. Pat. No. 4,862,160, U.S. Pat. No. 5,539,482, EP1466289,U.S. Pat. No. 5,268,834, U.S. Pat. No. 5,266,875, U.S. Pat. No.4,291,277, EP1423664, U.S. Pat. No. 6,611,823, WO 02/02974, WO02/095650, provide only optimal estimates for the model parameters ifthe model describes the behavior of the transducer completely. However,there are always differences between the theoretical model and the realtransducer which causes significant errors in the estimated nonlinearparameters (bias). This shall be described in the following section ingreater detail:

The output signal y(t) of the transducer:

y(t)=y _(nlin)(t)+y _(lin)(t)  (1)

consists of a nonlinear signal part:

y _(nlin)(t)=P _(sn) G _(n)(t)  (2)

and a linear signal part:

y _(lin)(t)=P _(sl) G _(l)(t)+e _(r)(t).  (3)

The linear signal part y_(lin)(t) comprises a scalar productP_(sl)G_(l)(t) of a linear parameter vector P_(sl), a gradient vectorG_(l)(t) and a residual signal e_(r)(t) due to measurement noise andimperfections of the model.

The nonlinear signal part y_(nlin)(t) can be interpreted as nonlineardistortion and can be described as a scalar product of the parametervector:

P_(sn)=[p_(s,1) p_(s,2) . . . p_(s,N)]  (4)

and the gradient vector:

G _(n) ^(T)(t)=[g ₁(t)g ₂(t) . . . g _(N)(t)]  (5)

which may contain, for example:

G _(n) ^(T)(t)=[i(t)x(t)² i(t)x(t)⁴ i(t)x ⁶]  (6)

products of input signal x(t) and the input current:

i(t)=h _(i)(t)*x(t).  (7)

The model generates an output signal:

y′(t)=P _(n) G _(n)(t)+P _(l) G _(l)(t),  (8)

which comprises scalar products of the nonlinear parameter vector:

P_(n)=[P_(n,1) P_(n,2) . . . P_(n,N)]  (9)

and the linear parameter vector:

P_(l)=[P_(l,1) P_(l,2) . . . P_(l,L)]  (10)

with the corresponding linear and nonlinear gradient vector G_(n)(t) andG_(l)(t), respectively. It is the target of the optimal systemidentification that the parameters of the model coincide with the trueparameters of the transducer (P_(n)→P_(s,n), P_(l)→P_(s,l)).

A suitable criterion for the agreement between model and reality is theerror time signal:

e(t)=y(t)−y′(t),  (11)

which can be represented as a sum:

e(t)=e _(n)(t)+e _(l)(t)+e _(r)(t)  (12)

comprising a nonlinear error part:

e _(n)(t)=ΔP _(n) G _(n)=(P _(sn) −P _(n))G _(n),  (13)

a linear error part:

e _(l)(t)=ΔP _(l) G _(l)(P _(sl) −P _(l))G _(l)  (14)

and the residual signal e_(r)(t).

System identification techniques known in the prior art determine thelinear and nonlinear parameters of the model by minimizing the totalerror e(t) in a cost function:

C=E{e(t)²}→Minimum.  (15)

The linear parameters P_(l) are estimated by inserting Eqs. (1) and (8)into Eq. (11), multiplying with the transposed gradient vector G_(l)^(T)(t) and calculating the expectation value

$\begin{matrix}{{{E\left\{ f \right\}} = {\lim\limits_{T\rightarrow\infty}{\left( {\frac{1}{T}{\int_{0}^{T}{f(t)}}} \right).}}}\ } & (16)\end{matrix}$

Considering that the residual error e_(r)(t) is not correlated with thelinear gradient signals in G_(l)(t), this results in theWiener-Hopf-equation:

P _(l) E{G _(l)(t)G _(l) ^(T)(t)}=E{y _(lin)(t)G _(l) ^(T)(t)}−E{e_(n)(t)G _(l) ^(T)(t)}P _(l) S _(GGl) =S _(yGl) +S _(rGl)  (17)

which can be solved directly by multiplying this equation with theinverted matrix S_(GGl):

P _(l)=(S _(yGl) +S _(rGl))S _(GGl) ⁻¹ P _(l) =P _(sl) +ΔP _(l)  (18)

or determined iteratively by using the LMS algorithm:

$\begin{matrix}\begin{matrix}{{P_{l}^{T}(t)} = {{P_{l}^{T}\left( {t - 1} \right)} + {\mu \; {G_{l}(t)}{e(t)}}}} \\{= \left. {{P_{l}^{T}\left( {t - 1} \right)} + {\mu \; {G_{l}(t)}{e_{l}(t)}} + {\mu \; {G_{l}(t)}{e_{n}(t)}}}\rightarrow \right.} \\{{P_{sl}^{T} + {\Delta \; P_{l}^{T}}}}\end{matrix} & (19)\end{matrix}$

with parameter μ changing the speed of convergence. The linearparameters P_(l) are estimated with a systematic bias ΔP_(l) if there isa correlation between the nonlinear error e_(n)(t) and the lineargradient vector G_(l)(t).

Minimizing the total error in the cost function in Eq. (15) may alsocause a systematic bias in the estimation of the nonlinear parametersP_(n). Inserting Eq. (1) and (8) into Eq. (11) and multiplying with thetransposed gradient vector G_(n) ^(T)(t) results in theWiener-Hopf-equation for the nonlinear parameters:

P _(n) E{G _(n)(t)G _(n) ^(T)(t)}=E{y _(nlin)(t)G _(n) ^(T)(t)}−E{[e_(l)(t)+e _(r)(t)]G _(n) ^(T)(t)}, P _(n) S _(GGn) =S _(yGn) =S _(yGn)+S _(rGn)  (20)

where S_(GGn) is the autocorrelation of the gradient signals, S_(yGn) isthe cross-correlation between the gradients and the signal y_(nlin)(t)and S_(rgn) is the cross-correlation of the residual error e_(r) withthe gradient signals. The nonlinear parameters of the model can directlybe calculated by inverting the matrix S_(GGn):

P _(n)=(S _(yGn) +S _(rGn))S _(GGn) ⁻¹ P _(n) =P _(sn) +ΔP _(n)  (20)

or iteratively by using the LMS-algorithm:

$\begin{matrix}\begin{matrix}{{P_{n}^{T}(t)} = {{P_{n}^{T}\left( {t - 1} \right)} + {\mu \; {G_{n}(t)}{e(t)}}}} \\{= \left. {{P_{n}^{T}\left( {t - 1} \right)} + {\mu \; {G_{n}(t)}{e_{n}(t)}} + {\mu \; {{G_{n}(t)}\left\lbrack {{e_{l}(t)} + {e_{r}(t)}} \right\rbrack}}}\rightarrow \right.} \\{{P_{sn}^{T} + {\Delta \; {P_{n}^{T}.}}}}\end{matrix} & (22)\end{matrix}$

These techniques known in prior art generate a systematic deviationΔP_(n) from the true parameter values if either the linear error e_(l)or the residual error e_(r) correlates with the nonlinear gradientG_(n):

E{G _(n)(t)[e _(l)(t)+e _(r)(t)]}≠0  (23)

The bias ΔP_(n) in the estimation of P_(n) is significant (>50%) if thenonlinear distortion y_(nlin) is small in comparison to the residualsignal e_(r)(t), which is mainly caused by imperfections in the linearmodeling.

To cope with this problem, the prior art increases the complexity of thelinear model (e.g. the number of taps in an FIR-filter) to describe thereal impulse response h_(m)(t) more completely. This demand can not berealized in many practical applications. For example, the suspension ina loudspeaker has a visco-elastic behavior which can hardly be modeledby a linear filter of reasonable order. The eddy currents induced in thepole plate of a loudspeaker also generate a high complexity of theelectrical input impedance. In addition, loudspeakers also behave astime varying systems where aging and changing ambient conditions(temperature, humidity) cause a mismatch between reality and model whichincreases the residual error signal e_(r)(t).

OBJECTS OF THE INVENTION

There is thus a need for an identification system which estimates thenonlinear parameters P_(n) and the linear parameters P_(l) of the modelwithout a systematic error (bias) if the measured signals are disturbedby noise or there are imperfections in the modeling of the transducer.The free parameters of the model should be identified by exciting thetransducer with a normal audio signal (e.g. music), a synthetic testsignal (e.g. noise) or a control signal as used in active noisecancellation having sufficient amplitude and bandwidth to providepersistent excitation. The transferred signal shall not or onlyminimally be changed by the identification system to avoid anydegradation of the subjectively perceived sound quality. A furtherobject is to realize an identification system for transducers comprisinga minimum of elements and requiring minimal processing capacity in adigital signal processor (DSP) to keep the cost of the system low.

SUMMARY OF THE INVENTION

According to the invention, the nonlinear parameters P_(n) are estimatedby minimizing the cost function:

C _(n) =E{e _(n)(t)²}→Minimum  (24)

which considers the nonlinear error part e_(n) only. In this case, thecorrelation:

E{G _(n)(t)e _(n)(t)}=0  (25)

between nonlinear error part e_(n)(t) and nonlinear gradient signalG_(n)(t) vanishes. Thus, a systematic error (bias) in the estimatedvalue of the nonlinear parameter P_(n) can be avoided.

The nonlinear cost function C_(n) is not suitable for an error-freeestimation of the linear parameters P_(l). Using different costfunctions for the estimation of the linear and nonlinear parameters is afeature of the current invention not found in prior art. Thisrequirement can be realized theoretically by splitting the total errore(t) into error components according to Eq. (12) and using only thenonlinear error part e_(n)(t) for the estimation of the nonlinearparameters P_(n). However, the practical realization is difficult and itis more advantageous to apply an appropriate transformation T_(g) to thegradient signal G_(n)(t) and to generate a modified gradient signal:

$\begin{matrix}\begin{matrix}{{G^{\prime \; T}(t)} = \left\lbrack {{g_{1}^{\prime}(t)}\mspace{31mu} {g_{2}^{\prime}(t)}\mspace{25mu} \cdots \mspace{25mu} {g_{N}^{\prime}(t)}} \right\rbrack} \\{= {T_{g}\left\{ {G_{n}^{T}(t)} \right\}}}\end{matrix} & (26)\end{matrix}$

and/or to transform the error signal e(t) by an appropriatetransformation T_(e) into a modified error signal:

e′ _(j)(t)=T _(e,j) {e(t)}=e′ _(n,j)(t)+e′ _(res,j)(t), j=1, . . . ,N.  (27)

The transformations T_(g) and T_(e) have to be chosen to ensure that thecorrelation:

E{g′ _(j)(t)e′ _(res,j)(t)}=0, j=1, . . . , N  (28)

between the transformed residual error e′_(res,j)(t) and the transformedgradient signals g′_(j)(t) will vanish, and a positive correlation:

E{g _(j)(t)g′ _(j)(t)^(T)}>0, j=1, . . . , N  (29)

between original and transformed gradient signals and a positivecorrelation:

E{e _(n)(t)e′ _(n,j)(t)}>0, j=1, . . . , N  (30)

between the original and transformed error is maintained.

The transformations T_(g) and T_(e) suppress the linear signal partsy_(nlin) primarily, but preserve most of the information of thenonlinear signal part y_(nlin) required for the estimation of thenonlinear parameters.

Using the transformed gradient signal g′(t) and the transformed errorsignal e′(t) in the LMS algorithm:

P _(n,j)(t)=p _(n,j)(t−1)+μg′ _(j)(t)e′ _(j)(t), j=1, . . . , NP_(n)→P_(sn)  (31)

results in an error-free estimation of the nonlinear parameters(P_(n)=P_(sn)). Suitable transformations can be realized by differentmethods:

The first method developed here is a new decorrelation technique whichhas the benefit that a modification of the input signal x(t) is notrequired. A signal with arbitrary temporal and spectral propertiesensuring persistent excitation of the transducer is supplied to thetransducer input 7. Although the decorrelation technique can be appliedto the output signal y(t), it is beneficial to calculate thedecorrelated error signal:

e′ _(j)(t)=T _(e,j) {e(t)}=e(t)+C _(j)B_(j), j=1, . . . , N  (32)

which is the sum of the original error signal and the jth compensationvector:

B_(j) ^(T)=[b_(j,l) b_(j,k) . . . b_(j,K)],  (33)

weighted by the jth decorrelation parameter vector:

C_(j)=[c_(j,l) c_(j,k) . . . c_(j,K)].  (34)

All compensation vectors B_(j) with j=1, . . . , N comprise onlydecorrelation signals b_(j,i) with i=1, . . . , K, which have a linearrelationship with the input signal x(t). Those decorrelation signalsb_(j,i) have to be derived from the transducer model and correspond withthe gradient signals g′_(j). The expectation value:

E{e′_(j)g_(j)}=ΣΠ{η_(k)η_(l)}  (35)

which is the product of the error signal e′_(j) and the gradient signalg_(j) can be decomposed into a sum of products in which each productcomprises only expectation values of two basic signals κ_(k) and κ_(l)(as described, for example, in “Average of the Product of GaussianVariables,” in M. Schertzen, “The Volterra and Wiener Theories ofNonlinear Systems”, Robert E. Krieger Publishing Company, Malabar, Fla.,1989.)

Applying Eq. (35) to the first gradient signal g_(l)(t)=ix² presented asan example in Eq. (6) results in:

E{ix ² e′ _(l)(t)}=2E{ix}E{xe _(l)′(t)}+E{ie _(l)′(t)}E{x ²}.  (36)

The correlation between the nonlinear gradient signal ix² and anarbitrary (linear) error part e′_(res,l)(t) in e′_(l)(t) vanishes if thefollowing conditions:

E{xe _(l)′(t)}=0E{ie _(l)′(t)}=0  (37)

hold.

The transformation T_(e,j) of the error signals e(t) has to remove thecorrelation between e′_(l)(t) and x and the correlation betweene′_(l)(t) and i as well. The compensation vector:

B_(l) ^(T)=[xi]  (38)

for j=1 comprises only displacement x(t) and current i(t) which areweighted by C_(j) and added to the original error signal e(t) asdecorrelation signals according to Eq. (32). The optimal decorrelationparameter C_(j) can be determined adaptively by the following iterativerelationship:

C _(j) ^(T)(t)=C _(j) ^(T)(t−1)+μB _(j) e′ _(j)(t), j=1, . . . ,N.  (39)

Using the additive decorrelation method the transformed gradient signalG′(t)=T_(g){G_(n)}=G_(n) is equal to the gradient signal G_(n). Thenonlinear information in the error part e_(n) which is required for theestimation of the nonlinear parameters P_(n) is preserved in thetransformed error signal e′_(j)(t). If the error signal e′_(j)(t)contains the nonlinear gradient signal e′_(n,l)(t)=ix², the expectationvalue:

E{ix ² ix ²}=2E{ii}E{xx} ²+4E{ix} ² E{xx}≠0,  (40)

will not vanish and the condition in Eq. (29) is fulfilled.

The transformed error signal e′_(i)(t) can not be used for theestimation of the linear parameters P_(l); the original error signale(t) according to Eq. (19) should be used instead.

An alternative transformation which fulfills the requirements of Eqs.(28)-(30) can be realized by performing a filtering:

x(t)=h _(g)(t)*u(t)  (41)

of the excitation signal with the filter function:

$\begin{matrix}{{H_{g}(f)} = {{{FT}\left\{ {h_{g}(t)} \right\}} = {\sum\limits_{i = 1}^{I}\left( {1 - {\delta \left( {f - f_{i}} \right)}} \right)}}} & (42)\end{matrix}$

where FT{ } is Fourier transformation and the function δ(f) is definedas:

$\begin{matrix}{{\delta \; (f)} = {\begin{Bmatrix}{1,{{{for}\mspace{14mu} f} = 0}} \\{0,{{{for}\mspace{14mu} f} \neq 0}}\end{Bmatrix}.}} & (43)\end{matrix}$

A few selected spectral components at frequencies f_(i) with i=1, . . .I do not pass the filter, but the remaining signal components aretransferred without attenuation.

A second filter with a transfer function:

$\begin{matrix}{{H_{a}(f)} = {{{FT}\left\{ {h_{a}(t)} \right\}} = {\sum\limits_{i = 1}^{I}\; {\delta \left( {f - f_{i}} \right)}}}} & (44)\end{matrix}$

is used for the transformation T_(e) of the error signal:

e′ _(j)(t)=h _(a)(t)*e _(j)(t), j=1, . . . , N.  (45)

Since the filter H_(a)(f) lets pass only spectral components which arenot in the input signal x(t), the transformed error signal e′_(j)(t)will not be correlated with the linear error signal e′_(res)(t)fulfilling the first condition in Eq. (28). However, the error signale′_(j)(t) contains sufficient nonlinear spectral components frome_(n,j), to ensure a correlation between both error signals according tothe second condition in Eq. (30).

The LMS algorithm applied to the filtered error signal:

p _(n,j)(t)=p _(n,j)(t−1)+μg _(j)(t)e′ _(j)(t), j=1, . . . , NP_(n)→P_(sn)  (46)

results in an error-free estimation of the nonlinear parameters as longas measurement noise is not correlated with the gradient signalG_(n)(t). If the error signal is filtered, the transformed gradientsignal G′(t)=T_(g){G_(n)}=G_(n) is identical with original gradientsignal.

A third alternative to realize the conditions in Eq. (28)-(30) is thefiltering of the gradient signals:

G′(t)=h _(a)(t)*G _(n)(t)  (47)

by using the filter function H_(a)(f) defined in Eq. (44) whilefiltering the input signal with the filter function H_(g)(f) accordingEq. (42).

This transformation ensures that the filtered gradient signal g′_(j)(t)is neither correlated with the linear error e_(r)(t) nor with theresidual error eat):

$\begin{matrix}{\left. \begin{matrix}{{E\left\{ {{g_{j}^{\prime}(t)}{e_{r}(t)}} \right\}} = 0} \\{{E\left\{ {{g_{j}^{\prime}(t)}{e_{l}(t)}} \right\}} = 0}\end{matrix} \right\},{j = 1},{\ldots \mspace{11mu} {N.}}} & (48)\end{matrix}$

Assuming that the measurement noise is not correlated with g′_(j)(t),the nonlinear parameters:

p _(n,j)(t)=p _(n,j)(t−1)+μg′ _(j)(t)e _(j)(t), j=1, . . . , NP_(n)→P_(sn)  (49)

can be estimated without bias using the LSM algorithm.

The total number of frequencies I and the values f_(i) with i=1, . . . ,I have to be selected in such a way to provide persistent excitation ofthe transducer and to get sufficient information from the nonlinearsystem. If the number I of frequencies is too large, the filtering ofthe input signal impairs the quality of transferred audio signal (music,speech).

The number I of the frequencies f_(i) can be significantly reduced(e.g., I=1) if the values of the frequencies are not constant but rathervary with a function f_(i)=f(t) of time. This extends the learning time,but causes only minimal changes in the transferred audio signal. When arelatively small number of frequencies I is used, it is not possible toidentify the order and contribution of each distortion component. Thisis a difference with respect to traditional methods used in prior artfor distortion measurements and nonlinear system identification.

If the nonlinear parameters P_(n) have been estimated without bias andthe nonlinear error e_(n)(t) disappears, the linear parameters P_(l) canbe estimated without bias by minimizing the cost function C in Eq. (15).

The current invention has the benefit that the linear and nonlinearparameters can be determined without a systematic error (bias), even ifthe modeling of the linear properties of the transducer is not perfect.This reduces the effort in modeling complicated mechanisms (e.g., creepof the suspension) and makes it possible to use models with lowercomplexity and a minimal number of free parameters. This is beneficialfor speeding up the identification process, improving the robustness andreducing implementation cost.

These and other features, aspects, and advantages of the presentinvention will become better understood with reference to the followingdrawings, description, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general block diagram showing a parameter identificationsystem for a measurement, diagnostics and control application known inthe prior art.

FIG. 2 is a general block diagram showing a parameter identificationsystem for a measurement, diagnostics and control application inaccordance with the present invention.

FIG. 3 shows a first embodiment of a transformation system using anadditive decorrelation technique.

FIG. 4 shows an alternative embodiment of the present invention whichuses a filter technique.

FIG. 5 shows an embodiment of a transformation system using an errorfilter in accordance with the present invention.

FIG. 6 shows an embodiment of a transformation system using a gradientfilter in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 is a general block diagram showing a parameter identificationsystem for measurement, diagnostics and control application in the priorart. The real transducer system 1 consists of a loudspeaker 3 (actuator)converting an electrical input signal x(t) (e.g., voltage at theterminals) at input 7 into a acoustical signal and a microphone 5(sensor) converting an acoustical signal into an electrical signal y(t)at output 9 which is supplied to the non-inverting input of an amplifier51. The transfer behavior of transducer system 1 is represented by Eq.(1). The input signal x(t) is also supplied via input 13 to a model 11.The model 11 describes the linear and nonlinear transfer behavior oftransducer system 1 and generates an output signal y′(t) at output 15which is supplied to the inverting input of the amplifier 51. Thenonlinear Eq. (8) describes the transfer behavior of the model 11,whereas the product of the linear parameter P_(l) and linear gradientvector G_(l)(t) is realized by using an FIR-filter. The nonlinear termP_(n)G_(n)(t) is realized by using linear filter, multiplier, adder andscaling elements according to Eqs. (6), (7) and (9). The error signale(t) generated at output 49 of the amplifier 51 according Eq. (11) issupplied to input 35 of the nonlinear parameter estimator 23 and toinput 31 of a linear parameter estimator 21. According to prior artdesigns, both parameter estimators 21 and 23 minimize the error signale(t) using the same cost function given in Eq. (15). The linearparameter estimator 21 is provided with the gradient vector G_(l) whichis generated in a linear gradient system 47 by using delay units andsupplied from output 37 to the input 29. Similarly the nonlineargradient system 41 generates the nonlinear gradient vector G_(n)according to Eq. (6) which is supplied via output 45 to the input 33 ofthe nonlinear parameter estimator 23. Both parameter estimators 21 and23 use the LMS algorithm as described in Eqs. (19) and (22). Both thelinear and the nonlinear gradient system 47 and 41 are supplied with theinput signal x(t) via inputs 39 and 43, respectively. The linearparameter vector P_(n) is generated at output 25 of the linear parameterestimator 21 and supplied to the input 19 of the model 11. The nonlinearparameter vector P_(n) is generated at output 27 of the nonlinearparameter estimator 23 and is supplied via input 17 to the model 11. Thelinear and nonlinear parameter vectors P_(l) and P_(a) are also suppliedto the diagnostic system 53 and to a controller 58, which is suppliedwith the control input z(t) and generates the input signal x(t) which issupplied to the transducer input 7. The controller 58 performs aprotection and linearization function for the transducer system 1. Ifthe model 11 describes the linear properties of the transducer system 1incompletely, the minimization of the cost function in Eq. (15) causes asystematic error (bias) in the estimation of the nonlinear parameterP_(n) as shown in Eqs. (21) and (22).

FIG. 2 is a block diagram showing a parameter identification system inaccordance with the invention, which avoids the bias in the estimationof the nonlinear parameters. The transducer system 1 comprisingloudspeaker 3 and microphone 5, model 11, the linear and nonlinearparameter estimators 21 and 23, respectively, the controller 58 and thediagnostic system 53 are identical with the corresponding elements shownin FIG. 1. The main difference to the prior art is that a transformationsystem 55 generates a modified error signal e′(t) and/or a modifiednonlinear gradient signal G′ which is supplied via outputs 67 and 69 tothe inputs 33 and 35, respectively, of the nonlinear parameter estimator23. The total error signal e(t) is transformed according T_(e) in Eq.(27) into the modified error signal e′(t). The nonlinear gradient vectorG_(n) is transformed according to T_(g) in Eq. (26) into the gradientvector G′. The special cost function C_(n) in Eq. (24) is used forestimating the nonlinear parameters P_(n) and the cost function C in Eq.(15) is used for the estimation of the linear parameters P_(l). Usingtwo different cost functions is a typical characteristic of the currentinvention. The transformation system 55 is supplied with the outputsignal y(t) from output 9 of the transducer system 1 via input 57 andwith the output signal y′(t) from output 15 of the model 11 via input59. The input signal x(t) from input 7 of the transducer system 1 isalso supplied to the input 61 of the transformation system 55.

FIG. 3 shows a first embodiment of the transformation system 55 using anadditive decorrelation technique. The transformation system 55 comprisesa linear gradient system 71, a nonlinear gradient system 87, anamplifier 85 and a decorrelation system 94. The linear and nonlineargradient systems 71 and 87 correspond with the gradient systems 47 and41 in FIG. 1, respectively. The input signal x(t) at input 61 issupplied to both the input of the linear gradient system 71 and theinput of the nonlinear gradient system 87. The output of the lineargradient system 71 is connected to an output 63 of the transformationsystem 55 at which the gradient vector G_(l)(t) is generated. Thegradient vector G′(t)=G_(n)(t) at the output of the nonlinear gradientsystem 87 is supplied to an output 69 of the transformation system 55.The linear gradient system 71 can be realized as a FIR-filter and thenonlinear gradient system 87 can be realized by using linear filters,multipliers, adders and scaling elements according Eq. (6). Thetransformation system 55 also includes an amplifier 85 similar to theamplifier 51 in FIG. 1. The modeled output signal y′(t) and the measuredsignal y(t) at inputs 59 and 57 of the transformation system 55 aresupplied to the inverting and non-inverting inputs of the amplifier 85,respectively. The error signal e(t) is generated at the output 101 ofthe amplifier 85 according to Eq. (11) and supplied to an output 65 ofthe transformation system 55 and to an error input 92 of thedecorrelation system 94. The decorrelation system 94 comprises asynthesis system 81, weighting elements 79 and 99, adders 83 and 75, amultiplier 77 and a storage element 73. The error signal e(t) at input92 is supplied to the scalar input of the adder 83. The adder 83 alsohas a vector input 84 provided with input signal C_(j)B_(j) and a vectoroutput 86 providing the output signal e_(j)′(t) with j=1, . . . , Naccording Eq. (32) to an output 67 of the transformation system 55. Thesignals in the jth compensation vector B_(j) with j=1, . . . , N aregenerated by using the synthesis system 81. The input 89 of synthesissystem 81 is connected to input 61 of the transformation system 55. Thesynthesis system 81 contains linear filters which may be realized bydigital signal processing. For each nonlinear gradient signal g_(j)(t),a set of decorrelation signals b_(j,i) in vector B_(j) is found bysplitting the expectation value E{g_(j)(t)e_(j)(t)} according to Eq.(36) in a sum of products. The output 91 of the synthesis system 81 isconnected to the input 95 of weighting element 79. The weighting element79 also has a vector input 93 provided with the decorrelation parametersC_(j) and an output 97 generating the weighted compensation signalC_(j)B_(j) supplied to input 84 of the adder 83. The optimaldecorrelation parameters C_(j) are generated adaptively according to Eq.(37). The transformed error signal e′_(j)(t) is supplied to a firstinput 96 of multiplier 77, and the compensation vector B_(j) is suppliedto the second (vector) input 98 of multiplier 77. The output signalB_(j)e′_(j)(t) of the multiplier 77 is supplied to an input of theweighting element 99 and is weighted by the learning constant μ. Theoutput signal μB_(j)e′_(j)(t) is added to the decorrelation parametervector C_(j) stored in the storage element 73 by using adder 75, and thesum is supplied to a control input 93 of the weighting element 79.

FIG. 4 shows a further embodiment of the invention using a filter 121which changes the spectral properties of the signal supplied to thetransducer system 1. The filter 121 has an input 119 supplied with theinput signal u(t) and an output 123 connected via controller 58 to theinput 7 of the transducer system 1. The filter 121 has a linear transferfunction according to Eq. (42). A few spectral components at frequenciesf_(i)=1, . . . I are suppressed while all the other components passthrough filter 121 without attenuation. The filter 121 can be realizedby using multiple filters with a band-stop characteristic which areconnected in series between filter input 119 and filter output 123.Alternatively, the filter 121 may be realized in a DSP by performing acomplex multiplication of the filter response H_(g)(f) with the inputsignal transformed into the frequency domain. The filter 121 may beequipped with an additional control input 117 connected with the outputof a frequency control system 115 to vary the frequencies f_(i) duringparameter identification. The frequency control system 115 can berealized as a simple oscillator generating a low frequency signalvarying the frequency f, of the band-stop filter for I=1. The output ofthe frequency control system 115 is also supplied to a control input 56of the transformation system 55.

FIG. 5 shows an embodiment of the transformation system 55 whichperforms filtering of the error signal. The transformation system 55contains a linear gradient system 71 and a nonlinear gradient system 87,having inputs connected with input 61 and having outputs providing thelinear gradient vector G_(l) and the nonlinear gradient vector G′=G_(n)to the outputs 63 and 69, respectively—similar to FIG. 3. Thetransformation system 55 in FIG. 5 also contains an amplifier 85 havinginverting and non-inverting inputs connected to inputs 57 and 59,respectively. The total error signal e(t) at output 101 is connected inthe same way as in FIG. 3 to output 65. An additional filter 105 havinga signal input 104 supplied with the output 101 of the amplifier 85 andhaving a filter output 103 generating the transformed error signal e′(t)has a linear transfer response according to Eq. (44) which can bechanged by the control signal provided via an input 106 fromtransformation system input 56. The filter 105 may be realized in thefrequency domain in a similar way as filter 121.

FIG. 6 shows an embodiment of the transformation system 55 whichperforms filtering of the gradient signals. The linear gradient system71, the nonlinear gradient system 87 and the amplifier 85 are connectedin the same way as described in FIG. 5. The total error signal e(t)supplied via output 101 to output 65 is identical with the transformederror signal e′(t) at vector output 67. The main difference to theprevious embodiments in FIGS. 3 and 4 is a filter 109 having a vectorinput 107 provided with nonlinear gradient vector G_(n) from the outputof the nonlinear gradient systems 87. The filter 109 has a transferfunction according to Eq. (44) which can be realized by a complexmultiplication in the frequency domain similar to the realization offilter 121. However, the filtering of the gradient signals generates ahigher computational load than the filtering of the error signal. Thetransformed gradient vector G′ at the vector output of filter 109 issupplied to output 69 of the transformation system 55. The transferbehavior of filter 109 may be varied by a control signal which isprovided via transformation system input 56 to an input 113 of filter109.

The embodiments of the invention described herein are exemplary andnumerous modifications, variations and rearrangements can be readilyenvisioned to achieve substantially equivalent results, all of which areintended to be embraced within the spirit and scope of the invention asdefined in the appended claims.

1. An arrangement for the optimal estimation of linear parameters P_(l)and nonlinear parameters P_(n) of a model 11, describing a transducer 1;said transducer 1 having at least one transducer input 7 which issupplied with an electrical, acoustical or arbitrary input signal x(t)and at least one transducer output 9 which generates an electrical,acoustical or arbitrary output signal y(t)=y_(lin)(t)+y_(nlin)(t), whichcontains a linear signal part y_(lin)(t) and a nonlinear signal party_(nlin)(t), said linear parameters P_(l) describing the linear signalpart y_(lin)(t) and the nonlinear parameters P_(nlin) describing thenonlinear signal part y_(nlin)(t) by using the model 11, comprising: atransformation system 55 having a first transformation input 61connected to receive said input signal x(t), a second transformationinput 57, connected to receive said output signal y(t), saidtransformation system arranged to produce a first transformation output67, 69 comprising time signals in which information on said linearsignal part y_(lin)(t) is suppressed and information on said nonlinearsignal part y_(nlin)(t) is preserved, and a second transformation output63, 65, in which information on the linear signal part y_(lin)(t) ispreserved, a nonlinear estimation system 23 having a signal input 33, 35connected to receive said first transformation output 67, 69 and havingan output 27 in which bias-free estimated nonlinear parameters P_(n) ofthe model 11 are generated even if the generation of the linear signalpart y_(lin)(t) by model 11 is incomplete, and a linear estimationsystem 21 having an input connected to receive said secondtransformation output 63, 65 and having an output 25 comprisingbias-free estimated linear parameters P_(l) of said model.
 2. Thearrangement of claim 1, wherein said nonlinear estimation system 23 isarranged to minimize a cost function C_(n), and said linear estimationsystem 21 is arranged to minimize a second cost function C, said costfunctions C_(n), and C being different from each other.
 3. Thearrangement of claim 2, wherein said transformation system 55 contains adecorrelation system 94 having a first decorrelation input 90 connectedwith the said first transformation input 61 and having a seconddecorrelation input 92 connected with said second transformation input57 and having a decorrelation output connected to said firsttransformation output 63, 65, and a signal with arbitrary temporal andspectral properties is supplied to the transducer input
 7. 4. Thearrangement of claim 3, wherein said decorrelation system 94 comprises:an adder 83 having multiple inputs and an output 86 providing the sum ofthe signals at the inputs, the first input of the adder provided withthe same signal as is supplied to said second decorrelation input 92, asynthesis system 81 having an input 89 connected with said firstdecorrelation input 90 and having multiple synthesis outputs 91providing linear decorrelation signals b_(j,i) which are derived fromthe decomposition of a nonlinear gradient signal g_(j), at least oneweighting element 79 having a first input 95 connected with one of thesynthesis outputs 91, each weighting element 79 having a second input 93provided with a decorrelation parameter c_(j,i) which is multiplied withsaid decorrelation signal b_(j,i) provided at the first input 95 andhaving an output 97 which generates the weighted decorrelation signalc_(j,i)b_(j,i) which is supplied to the inputs 84 of adders 83, and anestimation system for each decorrelation parameter c_(j,i) having afirst input 96 connected with the output 86 of the adder 83, having asecond input 98 connected with first input 95 of the correspondingweighting element 79 and having an output 65 generating saiddecorrelation parameter c_(j,i) which is supplied to said second input93 of said weighting element
 79. 5. The arrangement of claim 2, whereinsaid arrangement contains a first filter system 121 having a filterinput 119 provided with a signal u(t) with arbitrary temporal andspectral properties and having a filter output 123 generating a signalz(t) in which most of the spectral components of the input signals u(t)are preserved but a few spectral components at defined frequencies f_(i)with i=1, . . . , I are suppressed, said filter output 123 coupled withthe transducer input
 7. 6. The arrangement of claim 5, wherein saidtransformation system 55 contains a second filter system 105 having afilter input 104 coupled with said second transformation input 57, afilter output 103 providing only a few spectral components at the samefrequencies f_(i) with i=1, . . . , I as defined in the first filtersystem 121 while suppressing all other spectral components of the inputsignal provided to the filter input
 104. 7. The arrangement of claim 5,wherein said transformation system 55 comprising a nonlinear gradientsystem 87 having an input connected with the first transformation input61 and an output generating nonlinear gradient signals g_(j)(t) withj=1, . . . , N summarized in a vector G_(n)(t), a third filter system109 for each gradient signal g_(j)(t) with j=1, . . . , N, having aninput 107 connected with the corresponding output of the nonlineargradient system 87 and having a filter output 111 supplying atransformed gradient signal g_(j)′(t) with j=1, . . . N, totransformation system output 69 which comprises only a few spectralcomponents at frequencies f_(i) with i=1, . . . , I defined in the firstfilter system 121 but suppressing all other components with narrowpassbands.
 8. The arrangement of claim 5, wherein said first filtersystem 121 comprises a number I of band-stop filters connected in seriesbetween said filter input 119 and filter output 123 and each band-stopfilter i attenuates the spectral components at frequency f_(i) with i=1,. . . , I.
 9. The arrangement of claim 5, further comprising a frequencycontrol system 115 having an output connected to a control input 117 ofsaid first filter system 121, said output also connected to a controlinput 56 of said transformation system 55, said frequency control systemoutput causing said frequencies f_(i)(t) with i=1, . . . , I to varyversus time to estimate the linear and nonlinear parameters withoutbias.
 10. A method for estimating without bias the linear parametersP_(l) and nonlinear parameters P_(n) of a model 11 describing atransducer 1; said transducer 1 having at least one transducer input 7which is supplied with an electrical, acoustical or arbitrary inputsignal x(t) and at least one transducer output 9 which generates anelectrical, acoustical or arbitrary output signaly(t)=y_(lin)(t)+y_(nlin)(t) which contains a linear signal party_(lin)(t) and a nonlinear signal part y_(lin)(t), said linearparameters P_(l) describing the linear signal part y_(lin)(t) and saidnonlinear parameters P_(nlin) describing the nonlinear signal party_(lin)(t) by using the model 11, comprising the steps of: transformingsaid input signal x(t) and said output signal y(t) into firsttransformation signals G′ and e′, in which the linear signal party_(lin)(t) is suppressed and the information on the nonlinear signalpart y_(nlin)(t) is substantially preserved, transforming said inputsignal x(t) and said output signal y(t) into second transformationsignals G_(l) and e, in which information on said linear signal party_(lin)(t) is substantially preserved, estimating the nonlinearparameters P_(n) by minimizing a first cost function C_(n) and usingsaid first transformation signals G′ and e′, and estimating the linearparameters P_(l) by minimizing a second cost function C and using saidsecond transformation signals G_(l) and e.
 11. The method of claim 10,further comprising the steps of: calculating the predicted output y′(t)of the model 11, calculating the error signale(t)=y(t)−y′(t)=e_(l)+e_(n)+e_(r) which is the difference between saidoutput signal y(t) at the transducer output 9 and the predicted outputy′(t) of the model 11, said error signal e(t)=e_(l)+e_(n)+e_(r)comprising a linear error part e_(l), a nonlinear error part e_(n) and aresidual error part e_(r), calculating said first cost function C_(n) byconsidering the nonlinear error part e_(n) only, and calculating saidsecond cost function C by considering the total error signal e.
 12. Themethod of claim 11, further comprising the steps of: generating for eachnonlinear parameter P_(n,j) with j=1, . . . , N a decorrelation signalC_(j)B_(j), and adding said decorrelation signal C_(j)B_(j) to the errorsignal e(t) to generate the error signal e′(t) which is part of saidfirst transformation signal.
 13. The method of claim 12, furthercomprising the steps of: modeling the nonlinear signal part y_(n)(t) asa sum of gradient signals g_(j) weighted by the corresponding nonlinearparameter P_(n,j) with j=1, . . . , N, calculating an expectation valueE{e′_(j)g_(j)} of the product between gradient signal g_(j)(t) andtransformed error signal e′(t), splitting the expectation valueE{e′_(j)g_(j)} into a sum of products E{e_(j)′g_(j)}=ΣΠE{η_(k)η_(l)}, inwhich each partial product is the expectation value of two basic signalsη_(k) and η_(l) only, generating said basic signals η_(k) in the partialproduct E{e′_(j)η_(k)} by applying linear filtering to the input signalx(t), generating a compensation vector B_(j) for each nonlinearparameter P_(n,j) with j=1, . . . , N comprising said basic signalsη_(k) as decorrelation signals b_(j,k)=η_(k) with k=1, . . . , K, andgenerating said decorrelation signal C_(j)B_(j) for each nonlinearparameter P_(n,j) by multiplying the compensation vector B_(j) with thedecorrelation parameter vector C_(j).
 14. A method of claim 13, furthercomprising the step of: generating adaptively said decorrelationparameter vector C_(j) by using the decorrelated output signaly(t)+C_(j)B_(j) and the compensation vector B_(j).
 15. A method of claim11, further comprising the step of: applying a first linear filter withthe transfer function H_(p)(f) to an external input signal u(t) toattenuate a few spectral components in the input signal x(t) whereas thenumber M of transferred components is larger than the number I ofsuppressed components.
 16. A method of claim 15, further comprising thestep of: generating said first transformation signal by filtering theoutput signal y(t) with a transfer function H_(a)(f) which transfersspectral components which have been attenuated by the transfer functionH_(p)(f) in the first linear filter and attenuates spectral componentswhich have been transferred by the first filter.
 17. A method of claim15, further comprising the steps of: generating a nonlinear gradientsignal g_(j)(t) for each nonlinear parameter P_(n,j) with j=1, . . . , Nfrom the input signal x(t) by using the nonlinear modely_(nlin)(t)=P_(sn)G_(n) with G_(n) ^(T)(t)=[g₁(t) g₂(t) . . . g_(N)(t)],and generating said transformed gradient signal g_(j)′(t) by filteringeach gradient signal g_(j)(t) with a transfer function H_(a)(f) whichtransfers spectral components which have been attenuated by the transferfunction H_(p)(f) in the first linear filter and attenuates spectralcomponents which have been transferred by first filter.
 18. A method ofclaim 15, further comprising the step of: varying the properties of thetransfer response H_(p)(f) in the first filter during parameterestimation to change the frequencies of the spectral components whichhave been attenuated in the input signal x(t).